Finite Group Databasebeta

← #367009

Subgroups — #367009

Order 512.

2314 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2541C211yesmaximal, characteristic
2C2×#1982C211yesmaximal, characteristic, complemented
3C2×#1961C212yesmaximal, complemented
4C2×#1957C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic, complemented
6C2×#1770C211yesmaximal, characteristic, complemented
7C2×C4×#102C211yesmaximal, characteristic, complemented
8#13911C211yesmaximal, characteristic, complemented
9#11051C212yesmaximal, complemented
10#11008C212yesmaximal, complemented
11#10927C212yesmaximal, complemented
12C22×#127C2211yescharacteristic
13C22×#127C2212yes—
14C22×#121—22no—
15C22×C4×D4C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×#102C2211yescharacteristic
23C22×#102C2212yes—
24C22×#97C2212yes—
25C22×C4×C8C2211yescharacteristic
26#2541—24no—
27#2541C418yescomplemented
28C2×#392—22no—
29C2×#390C2212yes—
30C2×#384C2211yescharacteristic
31C2×#382C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1982—24no—
35#1961—28no—
36#1957—24no—
37#1802—24no—
38#1770—24no—
39C4×#102—24no—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2211yescharacteristic
43C2×#336C2212yes—
44#1442—22no—
45C4⋊#95—22no—
46C4⋊#95C2212yes—
47C23×D4—21no—
48C23×D4—24no—
49C23×D4—21no—
50C23×D4C2312yes—
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C23×C8C2311yescharacteristic
54C23×C8C2312yes—
55C23×C8C2311yescharacteristic
56C2×#127—24no—
57C2×#127—24no—
58C2×#127—22no—
59C2×#127—22no—
60C2×#127—24no—
61C2×#127—22no—
62C2×#127—22no—
63C2×#127—22no—
64C2×#127—42no—
65C2×#127—42no—
66C2×#127C2×C414yes—
67C2×#127C2×C418yes—
68C2×#121—216no—
69C2×#121—48no—
70C2×#121—48no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4C2×C414yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D412yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C22⋊C4—21no—
85C22×C22⋊C4—24no—
86C22×C42—21no—
87C22×C42C2312yes—
88C22×C42C2311yescharacteristic
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8D414yescomplemented
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102—42no—
108C2×#102—42no—
109C2×#102—42no—
110C2×#102—42no—
111C2×#102—42no—
112C2×#102D414yescomplemented
113C2×#97—24no—
114C2×#97—24no—
115C2×#97—42no—
116C2×C4×C8—22no—
117C2×C4×C8—22no—
118C2×C4×C8D414yescomplemented
119#392—216no—
120#390—24no—
121#390C2×C418yes—
122#384—22no—
123#384—22no—
124#382—24no—
125#382—42no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4C2×C414yes—
129C4×C4⋊C4D414yes—
130C2×#95—24no—
131C2×#95—21no—
132C2×#95—24no—
133C2×#95D412yes—
134#340—24no—
135#340—22no—
136#340—22no—
137#336—24no—
138C25—21no—
139C25—22no—
140C22×D4—24no—
141C22×D4—216no—
142C22×D4—24no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4—22no—
167C22×D4—48no—
168C22×D4—48no—
169C22×D4—48no—
170C22×D4—48no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—82no—
178C22×D4—82no—
179C22×D4—84no—
180C22×D4—84no—
181C22×D4C22×C414yes—
182C22×D4C22×C412yes—
183C22×D4C2×D412yes—
184C22×D4#4112yes—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4C2×D414yes—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2411yescharacteristic
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—22no—
214C22×C8C2×D412yes—
215C22×C8C2×D414yes—
216C22×C8C2×D412yes—
217C22×C8C2×D412yes—
218C22×C8#4112yes—
219#127—216no—
220#127—28no—
221#127—28no—
222#127—48no—
223#127—48no—
224#127—48no—
225#127—44no—
226#127—44no—
227#121—464no—
228C4×D4—28no—
229C4×D4—28no—
230C4×D4—44no—
231C4×D4—44no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—82no—
259C2×C4⋊C4C22⋊C418yes—
260C2×C4⋊C4C22×C412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4#4112yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—216no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—48no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—42no—
274C2×C42—44no—
275C2×C42C22×C414yes—
276C2×C42C22×C412yes—
277C2×C42C2×D414yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42#4112yes—
282C2×C42#4112yes—
283C4⋊C8—28no—
284#102—28no—
285#102—28no—
286#102—48no—
287#102—48no—
288#102—48no—
289#102—48no—
290#102—48no—
291#97—48no—
292C4×C8—28no—
293#95—48no—
294#95—42no—
295#95—84no—
296#95C22⋊C418yes—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—24no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—44no—
316C24—44no—
317C24—44no—
318C24—44no—
319C24C22×D411yescharacteristic
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—22no—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—432no—
342C2×D4—432no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—41no—
352C2×D4—42no—
353C2×D4—41no—
354C2×D4—42no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4—41no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—41no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—41no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—41no—
376C2×D4—42no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—416no—
380C2×D4—88no—
381C2×D4—816no—
382C2×D4—88no—
383C2×D4—84no—
384C2×D4—84no—
385C2×D4—816no—
386C2×D4—816no—
387C2×D4—816no—
388C2×D4—816no—
389C2×D4—88no—
390C2×D4—84no—
391C2×D4—84no—
392C2×D4C4×D414yes—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—48no—
417C22×C4—416no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—88no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4C2×C22⋊C418yes—
427C22×C4C2×C22⋊C412yes—
428C22×C4C2×C22⋊C412yes—
429C22×C4#12412yes—
430C22×C4#12412yes—
431C22×C4#12712yes—
432C22×C4#12712yes—
433C22×C4AGL(1,ℤ/8)14yes—
434C22×C4AGL(1,ℤ/8)14yes—
435C22×C4C23×C411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4H(5,𝔽2)11yescharacteristic
446C22×C4H(5,𝔽2)11yescharacteristic
447C22×C4H(5,𝔽2)11yescharacteristic
448C22×C42−1+411yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—42no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8#12114yescomplemented
470C4⋊C4—216no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C22⋊C4—432no—
490C22⋊C4—48no—
491C42—24no—
492C42—24no—
493C42—24no—
494C42—24no—
495C42—84no—
496C42C4×D414yes—
497C42C4×D414yes—
498C42#12114yes—
499C42#12114yes—
500C23—24no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—48no—
516C23—48no—
517C23—44no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23—81no—
525C23—81no—
526C23—81no—
527C23—81no—
528C23—84no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23—84no—
534C23—82no—
535C23—82no—
536C23—82no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—84no—
541C23—84no—
542C23—82no—
543C23—82no—
544C23—84no—
545C23—84no—
546C23C22×C22⋊C411yescharacteristic
547C23C2×#12411yescharacteristic
548C23C2×#12711yescharacteristic
549C23D4211yescharacteristic
550C23#54611yescharacteristic
551C23#54811yescharacteristic
552C23#55011yescharacteristic
553C23C2×AGL(1,ℤ/8)12yes—
554C23C2×AGL(1,ℤ/8)12yes—
555C23#57612yes—
556C23#57712yes—
557D4—24no—
558D4—24no—
559D4—42no—
560D4—44no—
561D4—44no—
562D4—42no—
563D4—44no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—44no—
569D4—816no—
570D4—864no—
571D4—82no—
572D4—82no—
573D4—82no—
574D4—82no—
575D4—816no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—82no—
602C2×C4—816no—
603C2×C4—84no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—84no—
607C2×C4—82no—
608C2×C4#39014yes—
609C2×C4#39014yes—
610C2×C4#41818yes—
611C2×C4#49018yes—
612C2×C4#49614yes—
613C2×C4#49614yes—
614C2×C4C2×C4×D411yescharacteristic
615C2×C4C2×C4×D411yescharacteristic
616C2×C4C2×C4×D411yescharacteristic
617C2×C4C4×#4111yescharacteristic
618C2×C4#51811yescharacteristic
619C2×C4#52011yescharacteristic
620C2×C4C2×#12111yescharacteristic
621C2×C4C2×#12111yescharacteristic
622C2×C4C2×#12111yescharacteristic
623C2×C4#52511yescharacteristic
624C2×C4#52911yescharacteristic
625C2×C4#53211yescharacteristic
626C2×C4#53511yescharacteristic
627C2×C4#53711yescharacteristic
628C2×C4#54011yescharacteristic
629C2×C4#54211yescharacteristic
630C2×C4D4211yescharacteristic
631C2×C4D4211yescharacteristic
632C2×C4#54611yescharacteristic
633C2×C4#54611yescharacteristic
634C2×C4#54711yescharacteristic
635C2×C4#54811yescharacteristic
636C2×C4#54811yescharacteristic
637C2×C4#55011yescharacteristic
638C2×C4#55011yescharacteristic
639C2×C4#55311yescharacteristic
640C2×C4#55311yescharacteristic
641C2×C4#55511yescharacteristic
642C8—24no—
643C8—42no—
644C8—44no—
645C22—48no—
646C22—44no—
647C22—82no—
648C22—84no—
649C22—82no—
650C22—82no—
651C22—84no—
652C22—84no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—88no—
657C22—88no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—162no—
662C22—162no—
663C22—162no—
664C22—164no—
665C22—164no—
666C22—162no—
667C22C2×#39011yescharacteristic
668C22D4×C22⋊C411yescharacteristic
669C22#234111yescharacteristic
670C22#249011yescharacteristic
671C22#249111yescharacteristic
672C22#254011yescharacteristic
673C22#254111yescharacteristic
674C22C2×#41812yes—
675C22#289712yes—
676C22C2×#49012yes—
677C22#313612yes—
678C22C2×#49611yescharacteristic
679C22C2×#49611yescharacteristic
680C22#315411yescharacteristic
681C22#315511yescharacteristic
682C22#327212yes—
683C22#327412yes—
684C22#332412yes—
685C22#332512yes—
686C22#336612yes—
687C22#336812yes—
688C22#339712yes—
689C22#339912yes—
690C4—24no—
691C4—48no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—162no—
696C4#196118yescomplemented
697C4C4×#12111yescharacteristic
698C4#232411yescharacteristic
699C4#236111yescharacteristic
700C4#236311yescharacteristic
701C4#241411yescharacteristic
702C4#241611yescharacteristic
703C4#242611yescharacteristic
704C4#242811yescharacteristic
705C2—84no—
706C2—164no—
707C2#1399811yescharacteristic
708C2C2×#196111yescharacteristic
709C2#2120611yescharacteristic
710C2#2242812yes—
711C2#2254512yes—
712C2#2775512yes—
713C2#2777512yes—
714C2#2841311yescharacteristic
715C2#2842011yescharacteristic
716C2#2847011yescharacteristic
717C2#2847211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2541C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1982C2semidirect
3. C2×#1961C2semidirect
4. C2×#1957C2semidirect
5. C2×#1802C2semidirect
6. C2×#1770C2semidirect
7. C2×C4×#102C2semidirect
8. #13911C2semidirect
9. #11051C2semidirect
10. #11008C2semidirect
11. #10927C2semidirect
27. #2541C4semidirect
91. C2×C4⋊C8D4semidirect
112. C2×#102D4semidirect
118. C2×C4×C8D4semidirect
469. C2×C8#121semidirect
696. C4#1961semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#25412
C2×#19822
C2×#19612
C2×#19572
C2×#18022
C2×#17702
C2×C4×#1022
#139112
#110512
#110082
#109272